General coding theorems for turbo-like codes

In this paper we prove that for general memoryless binary input channels, most ensembles of parallel and serial turbo codes, with fixed component codes, are "good" in the sense that with maximum likelihood decoding, their word (or bit) error probability decreases to zero as the block length increases, provided the noise is below a finite threshold. Our proof uses the classical union bound, which shows that under very general conditions, if the noise is below a certain threshold, the word (or bit) error probability is controlled by the low-weight codewords as the block length approaches infinity. Our main coding theorems then follow from a study of the low weight terms in the ensemble weight enumerator. Using this methodology, we can prove that the threshold is finite for most ensembles of parallel and serial turbo codes.